Yes, Carnap's rule (also known as Carnap rule) deduces \forall n A(n)
from the infinite number of premises A(0), A(1), ... Youc anf ind
numerous papers by Googling, I first learn about it from G. Mints'
lectures.
It can formalized if we have a recursive infinite tree.
> From Ron Rood
> In a Hilbert or Gentzen-style logic, logical derivations are usually
> represented as finite sequences of sentences.
> Are there studies of logics in the style of Hilbert or Gentzen
(natural
> deduction) which, in one way or another, allow for logical derivations
> of infinite length? If yes, can you provide me with pointers to the
> literature?